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Properties

Label 42.0.123...863.1
Degree $42$
Signature $[0, 21]$
Discriminant $-1.236\times 10^{87}$
Root discriminant \(118.47\)
Ramified primes $3,29$
Class number not computed
Class group not computed
Galois group $C_{42}$ (as 42T1)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^42 + 6*x^40 - 4*x^39 - 207*x^38 - 72*x^37 - 857*x^36 + 225*x^35 + 13524*x^34 + 4498*x^33 + 47646*x^32 - 1779*x^31 - 319728*x^30 + 56511*x^29 - 873558*x^28 + 1268358*x^27 + 3182931*x^26 - 287550*x^25 + 12585242*x^24 - 25238916*x^23 + 37767705*x^22 + 35532417*x^21 + 14671665*x^20 + 387146580*x^19 - 33604015*x^18 + 83426931*x^17 + 893018691*x^16 - 978644267*x^15 + 2536898589*x^14 + 320653281*x^13 + 1627927205*x^12 + 1198131426*x^11 - 1595121042*x^10 - 2227675498*x^9 - 1701355491*x^8 - 2415610293*x^7 - 905477771*x^6 + 53305794*x^5 + 1490523864*x^4 + 2070850257*x^3 + 1799361396*x^2 + 662562267*x + 198529417)
 
gp: K = bnfinit(y^42 + 6*y^40 - 4*y^39 - 207*y^38 - 72*y^37 - 857*y^36 + 225*y^35 + 13524*y^34 + 4498*y^33 + 47646*y^32 - 1779*y^31 - 319728*y^30 + 56511*y^29 - 873558*y^28 + 1268358*y^27 + 3182931*y^26 - 287550*y^25 + 12585242*y^24 - 25238916*y^23 + 37767705*y^22 + 35532417*y^21 + 14671665*y^20 + 387146580*y^19 - 33604015*y^18 + 83426931*y^17 + 893018691*y^16 - 978644267*y^15 + 2536898589*y^14 + 320653281*y^13 + 1627927205*y^12 + 1198131426*y^11 - 1595121042*y^10 - 2227675498*y^9 - 1701355491*y^8 - 2415610293*y^7 - 905477771*y^6 + 53305794*y^5 + 1490523864*y^4 + 2070850257*y^3 + 1799361396*y^2 + 662562267*y + 198529417, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^42 + 6*x^40 - 4*x^39 - 207*x^38 - 72*x^37 - 857*x^36 + 225*x^35 + 13524*x^34 + 4498*x^33 + 47646*x^32 - 1779*x^31 - 319728*x^30 + 56511*x^29 - 873558*x^28 + 1268358*x^27 + 3182931*x^26 - 287550*x^25 + 12585242*x^24 - 25238916*x^23 + 37767705*x^22 + 35532417*x^21 + 14671665*x^20 + 387146580*x^19 - 33604015*x^18 + 83426931*x^17 + 893018691*x^16 - 978644267*x^15 + 2536898589*x^14 + 320653281*x^13 + 1627927205*x^12 + 1198131426*x^11 - 1595121042*x^10 - 2227675498*x^9 - 1701355491*x^8 - 2415610293*x^7 - 905477771*x^6 + 53305794*x^5 + 1490523864*x^4 + 2070850257*x^3 + 1799361396*x^2 + 662562267*x + 198529417);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^42 + 6*x^40 - 4*x^39 - 207*x^38 - 72*x^37 - 857*x^36 + 225*x^35 + 13524*x^34 + 4498*x^33 + 47646*x^32 - 1779*x^31 - 319728*x^30 + 56511*x^29 - 873558*x^28 + 1268358*x^27 + 3182931*x^26 - 287550*x^25 + 12585242*x^24 - 25238916*x^23 + 37767705*x^22 + 35532417*x^21 + 14671665*x^20 + 387146580*x^19 - 33604015*x^18 + 83426931*x^17 + 893018691*x^16 - 978644267*x^15 + 2536898589*x^14 + 320653281*x^13 + 1627927205*x^12 + 1198131426*x^11 - 1595121042*x^10 - 2227675498*x^9 - 1701355491*x^8 - 2415610293*x^7 - 905477771*x^6 + 53305794*x^5 + 1490523864*x^4 + 2070850257*x^3 + 1799361396*x^2 + 662562267*x + 198529417)
 

\( x^{42} + 6 x^{40} - 4 x^{39} - 207 x^{38} - 72 x^{37} - 857 x^{36} + 225 x^{35} + 13524 x^{34} + \cdots + 198529417 \) Copy content Toggle raw display

sage: K.defining_polynomial()
 
gp: K.pol
 
magma: DefiningPolynomial(K);
 
oscar: defining_polynomial(K)
 

Invariants

Degree:  $42$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[0, 21]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(-123\!\cdots\!863\) \(\medspace = -\,3^{63}\cdot 29^{39}\) Copy content Toggle raw display
sage: K.disc()
 
gp: K.disc
 
magma: OK := Integers(K); Discriminant(OK);
 
oscar: OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(118.47\)
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(OK))^(1/Degree(K));
 
oscar: (1.0 * dK)^(1/degree(K))
 
Galois root discriminant:  $3^{3/2}29^{13/14}\approx 118.47395320948151$
Ramified primes:   \(3\), \(29\) Copy content Toggle raw display
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(OK));
 
oscar: prime_divisors(discriminant((OK)))
 
Discriminant root field:  \(\Q(\sqrt{-87}) \)
$\card{ \Gal(K/\Q) }$:  $42$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is Galois and abelian over $\Q$.
Conductor:  \(261=3^{2}\cdot 29\)
Dirichlet character group:    $\lbrace$$\chi_{261}(256,·)$, $\chi_{261}(1,·)$, $\chi_{261}(260,·)$, $\chi_{261}(5,·)$, $\chi_{261}(7,·)$, $\chi_{261}(136,·)$, $\chi_{261}(139,·)$, $\chi_{261}(16,·)$, $\chi_{261}(149,·)$, $\chi_{261}(25,·)$, $\chi_{261}(158,·)$, $\chi_{261}(35,·)$, $\chi_{261}(38,·)$, $\chi_{261}(167,·)$, $\chi_{261}(169,·)$, $\chi_{261}(71,·)$, $\chi_{261}(173,·)$, $\chi_{261}(175,·)$, $\chi_{261}(49,·)$, $\chi_{261}(179,·)$, $\chi_{261}(52,·)$, $\chi_{261}(181,·)$, $\chi_{261}(62,·)$, $\chi_{261}(199,·)$, $\chi_{261}(190,·)$, $\chi_{261}(80,·)$, $\chi_{261}(209,·)$, $\chi_{261}(82,·)$, $\chi_{261}(212,·)$, $\chi_{261}(86,·)$, $\chi_{261}(88,·)$, $\chi_{261}(92,·)$, $\chi_{261}(94,·)$, $\chi_{261}(223,·)$, $\chi_{261}(226,·)$, $\chi_{261}(103,·)$, $\chi_{261}(236,·)$, $\chi_{261}(112,·)$, $\chi_{261}(245,·)$, $\chi_{261}(122,·)$, $\chi_{261}(125,·)$, $\chi_{261}(254,·)$$\rbrace$
This is a CM field.
Reflex fields:  unavailable$^{1048576}$

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{17}a^{16}+\frac{8}{17}a^{15}-\frac{4}{17}a^{14}+\frac{2}{17}a^{13}-\frac{1}{17}a^{12}-\frac{8}{17}a^{11}+\frac{4}{17}a^{10}-\frac{2}{17}a^{9}+\frac{1}{17}a^{8}+\frac{8}{17}a^{7}-\frac{4}{17}a^{6}+\frac{2}{17}a^{5}-\frac{1}{17}a^{4}-\frac{8}{17}a^{3}+\frac{4}{17}a^{2}-\frac{2}{17}a$, $\frac{1}{17}a^{17}-\frac{1}{17}a$, $\frac{1}{17}a^{18}-\frac{1}{17}a^{2}$, $\frac{1}{17}a^{19}-\frac{1}{17}a^{3}$, $\frac{1}{17}a^{20}-\frac{1}{17}a^{4}$, $\frac{1}{17}a^{21}-\frac{1}{17}a^{5}$, $\frac{1}{17}a^{22}-\frac{1}{17}a^{6}$, $\frac{1}{17}a^{23}-\frac{1}{17}a^{7}$, $\frac{1}{17}a^{24}-\frac{1}{17}a^{8}$, $\frac{1}{17}a^{25}-\frac{1}{17}a^{9}$, $\frac{1}{17}a^{26}-\frac{1}{17}a^{10}$, $\frac{1}{17}a^{27}-\frac{1}{17}a^{11}$, $\frac{1}{17}a^{28}-\frac{1}{17}a^{12}$, $\frac{1}{289}a^{29}-\frac{7}{289}a^{28}-\frac{4}{289}a^{27}+\frac{5}{289}a^{26}+\frac{1}{289}a^{25}+\frac{3}{289}a^{24}-\frac{6}{289}a^{23}-\frac{7}{289}a^{22}+\frac{4}{289}a^{21}-\frac{3}{289}a^{20}-\frac{7}{289}a^{19}-\frac{8}{289}a^{18}-\frac{3}{289}a^{17}-\frac{7}{289}a^{16}+\frac{12}{289}a^{15}-\frac{125}{289}a^{14}-\frac{83}{289}a^{13}-\frac{3}{289}a^{12}+\frac{43}{289}a^{11}+\frac{69}{289}a^{10}-\frac{21}{289}a^{9}+\frac{109}{289}a^{8}-\frac{135}{289}a^{7}+\frac{120}{289}a^{6}-\frac{103}{289}a^{5}-\frac{109}{289}a^{4}+\frac{12}{289}a^{3}-\frac{122}{289}a^{2}-\frac{5}{17}a$, $\frac{1}{289}a^{30}-\frac{2}{289}a^{28}-\frac{6}{289}a^{27}+\frac{2}{289}a^{26}-\frac{7}{289}a^{25}-\frac{2}{289}a^{24}+\frac{2}{289}a^{23}+\frac{6}{289}a^{22}+\frac{8}{289}a^{21}+\frac{6}{289}a^{20}-\frac{6}{289}a^{19}-\frac{8}{289}a^{18}+\frac{6}{289}a^{17}-\frac{3}{289}a^{16}-\frac{58}{289}a^{15}+\frac{62}{289}a^{14}+\frac{62}{289}a^{13}-\frac{63}{289}a^{12}+\frac{81}{289}a^{11}+\frac{54}{289}a^{10}-\frac{89}{289}a^{9}+\frac{101}{289}a^{8}-\frac{26}{289}a^{7}-\frac{28}{289}a^{6}+\frac{122}{289}a^{5}+\frac{48}{289}a^{4}-\frac{72}{289}a^{3}+\frac{13}{289}a^{2}-\frac{7}{17}a$, $\frac{1}{289}a^{31}-\frac{3}{289}a^{28}-\frac{6}{289}a^{27}+\frac{3}{289}a^{26}+\frac{8}{289}a^{24}-\frac{6}{289}a^{23}-\frac{6}{289}a^{22}-\frac{3}{289}a^{21}+\frac{5}{289}a^{20}-\frac{5}{289}a^{19}+\frac{7}{289}a^{18}+\frac{8}{289}a^{17}-\frac{4}{289}a^{16}+\frac{52}{289}a^{15}+\frac{118}{289}a^{14}-\frac{93}{289}a^{13}-\frac{10}{289}a^{12}-\frac{115}{289}a^{11}+\frac{32}{289}a^{10}-\frac{77}{289}a^{9}-\frac{29}{289}a^{8}-\frac{43}{289}a^{7}+\frac{90}{289}a^{6}-\frac{5}{289}a^{5}-\frac{86}{289}a^{4}+\frac{54}{289}a^{3}-\frac{108}{289}a^{2}-\frac{2}{17}a$, $\frac{1}{289}a^{32}+\frac{7}{289}a^{28}+\frac{8}{289}a^{27}-\frac{2}{289}a^{26}-\frac{6}{289}a^{25}+\frac{3}{289}a^{24}-\frac{7}{289}a^{23}-\frac{7}{289}a^{22}+\frac{3}{289}a^{20}+\frac{3}{289}a^{19}+\frac{1}{289}a^{18}+\frac{4}{289}a^{17}-\frac{3}{289}a^{16}-\frac{118}{289}a^{15}-\frac{43}{289}a^{14}-\frac{38}{289}a^{13}-\frac{124}{289}a^{12}+\frac{127}{289}a^{11}+\frac{11}{289}a^{10}-\frac{7}{289}a^{9}-\frac{39}{289}a^{8}-\frac{26}{289}a^{7}-\frac{104}{289}a^{6}+\frac{132}{289}a^{5}+\frac{33}{289}a^{4}-\frac{106}{289}a^{3}+\frac{25}{289}a^{2}+\frac{5}{17}a$, $\frac{1}{289}a^{33}+\frac{6}{289}a^{28}-\frac{8}{289}a^{27}-\frac{7}{289}a^{26}-\frac{4}{289}a^{25}+\frac{6}{289}a^{24}+\frac{1}{289}a^{23}-\frac{2}{289}a^{22}-\frac{8}{289}a^{21}+\frac{7}{289}a^{20}-\frac{1}{289}a^{19}-\frac{8}{289}a^{18}+\frac{1}{289}a^{17}-\frac{1}{289}a^{16}+\frac{128}{289}a^{15}-\frac{13}{289}a^{14}+\frac{15}{289}a^{13}+\frac{131}{289}a^{12}+\frac{67}{289}a^{11}+\frac{37}{289}a^{10}-\frac{28}{289}a^{9}+\frac{112}{289}a^{8}-\frac{26}{289}a^{7}-\frac{62}{289}a^{6}+\frac{6}{289}a^{5}+\frac{28}{289}a^{4}+\frac{26}{289}a^{3}+\frac{123}{289}a^{2}-\frac{6}{17}a$, $\frac{1}{4913}a^{34}-\frac{4}{4913}a^{33}+\frac{4}{4913}a^{31}+\frac{1}{4913}a^{30}-\frac{3}{4913}a^{29}+\frac{1}{289}a^{28}+\frac{82}{4913}a^{27}+\frac{129}{4913}a^{26}-\frac{96}{4913}a^{25}+\frac{116}{4913}a^{24}+\frac{26}{4913}a^{23}+\frac{45}{4913}a^{22}-\frac{35}{4913}a^{21}+\frac{58}{4913}a^{20}-\frac{120}{4913}a^{19}-\frac{62}{4913}a^{18}+\frac{128}{4913}a^{17}+\frac{23}{4913}a^{16}+\frac{1472}{4913}a^{15}-\frac{552}{4913}a^{14}-\frac{665}{4913}a^{13}+\frac{1065}{4913}a^{12}-\frac{1269}{4913}a^{11}-\frac{496}{4913}a^{10}+\frac{2447}{4913}a^{9}+\frac{2287}{4913}a^{8}+\frac{1280}{4913}a^{7}+\frac{696}{4913}a^{6}+\frac{761}{4913}a^{5}+\frac{718}{4913}a^{4}-\frac{13}{4913}a^{3}-\frac{139}{289}a^{2}-\frac{8}{17}a$, $\frac{1}{4913}a^{35}+\frac{1}{4913}a^{33}+\frac{4}{4913}a^{32}+\frac{1}{4913}a^{30}+\frac{5}{4913}a^{29}+\frac{14}{4913}a^{28}+\frac{134}{4913}a^{27}-\frac{39}{4913}a^{26}-\frac{47}{4913}a^{25}-\frac{122}{4913}a^{24}-\frac{21}{4913}a^{23}-\frac{76}{4913}a^{22}+\frac{122}{4913}a^{21}-\frac{143}{4913}a^{20}+\frac{104}{4913}a^{19}-\frac{86}{4913}a^{18}+\frac{127}{4913}a^{17}-\frac{7}{289}a^{16}-\frac{2331}{4913}a^{15}+\frac{108}{289}a^{14}+\frac{1686}{4913}a^{13}-\frac{2415}{4913}a^{12}+\frac{1857}{4913}a^{11}-\frac{1186}{4913}a^{10}+\frac{1348}{4913}a^{9}+\frac{1843}{4913}a^{8}+\frac{2348}{4913}a^{7}-\frac{1640}{4913}a^{6}+\frac{192}{4913}a^{5}+\frac{1907}{4913}a^{4}+\frac{577}{4913}a^{3}-\frac{19}{289}a^{2}-\frac{6}{17}a$, $\frac{1}{4913}a^{36}+\frac{8}{4913}a^{33}-\frac{3}{4913}a^{31}+\frac{4}{4913}a^{30}-\frac{53}{4913}a^{28}-\frac{53}{4913}a^{27}+\frac{28}{4913}a^{26}-\frac{43}{4913}a^{25}+\frac{101}{4913}a^{24}-\frac{93}{4913}a^{22}+\frac{113}{4913}a^{21}+\frac{97}{4913}a^{20}-\frac{8}{289}a^{19}+\frac{36}{4913}a^{18}+\frac{93}{4913}a^{17}+\frac{77}{4913}a^{16}-\frac{996}{4913}a^{15}+\frac{28}{4913}a^{14}-\frac{628}{4913}a^{13}-\frac{1180}{4913}a^{12}+\frac{508}{4913}a^{11}-\frac{196}{4913}a^{10}+\frac{42}{4913}a^{9}+\frac{231}{4913}a^{8}-\frac{1781}{4913}a^{7}-\frac{1677}{4913}a^{6}+\frac{2319}{4913}a^{5}-\frac{600}{4913}a^{4}+\frac{931}{4913}a^{3}+\frac{142}{289}a^{2}-\frac{4}{17}a$, $\frac{1}{4913}a^{37}-\frac{2}{4913}a^{33}-\frac{3}{4913}a^{32}+\frac{6}{4913}a^{31}-\frac{8}{4913}a^{30}+\frac{5}{4913}a^{29}+\frac{134}{4913}a^{28}-\frac{118}{4913}a^{27}+\frac{13}{4913}a^{26}-\frac{117}{4913}a^{25}+\frac{109}{4913}a^{24}+\frac{124}{4913}a^{23}-\frac{43}{4913}a^{22}+\frac{105}{4913}a^{21}+\frac{97}{4913}a^{20}+\frac{44}{4913}a^{19}-\frac{40}{4913}a^{18}+\frac{56}{4913}a^{17}-\frac{75}{4913}a^{16}-\frac{2364}{4913}a^{15}-\frac{1788}{4913}a^{14}+\frac{536}{4913}a^{13}-\frac{481}{4913}a^{12}-\frac{1995}{4913}a^{11}+\frac{1562}{4913}a^{10}+\frac{1106}{4913}a^{9}-\frac{55}{289}a^{8}-\frac{87}{289}a^{7}-\frac{359}{4913}a^{6}-\frac{2183}{4913}a^{5}+\frac{32}{4913}a^{4}-\frac{2191}{4913}a^{3}-\frac{138}{289}a^{2}-\frac{2}{17}a$, $\frac{1}{5212693}a^{38}-\frac{308}{5212693}a^{37}-\frac{139}{5212693}a^{36}-\frac{372}{5212693}a^{35}-\frac{208}{5212693}a^{34}-\frac{1866}{5212693}a^{33}-\frac{8667}{5212693}a^{32}-\frac{4371}{5212693}a^{31}+\frac{2202}{5212693}a^{30}+\frac{2741}{5212693}a^{29}-\frac{101060}{5212693}a^{28}+\frac{145199}{5212693}a^{27}-\frac{27440}{5212693}a^{26}-\frac{74604}{5212693}a^{25}+\frac{52507}{5212693}a^{24}+\frac{129818}{5212693}a^{23}+\frac{3730}{5212693}a^{22}+\frac{16420}{5212693}a^{21}+\frac{131264}{5212693}a^{20}+\frac{67368}{5212693}a^{19}+\frac{130472}{5212693}a^{18}-\frac{98609}{5212693}a^{17}+\frac{88034}{5212693}a^{16}+\frac{229828}{5212693}a^{15}+\frac{228563}{5212693}a^{14}-\frac{1079584}{5212693}a^{13}-\frac{1981247}{5212693}a^{12}+\frac{617282}{5212693}a^{11}-\frac{928477}{5212693}a^{10}-\frac{199356}{5212693}a^{9}+\frac{149824}{306629}a^{8}-\frac{2241674}{5212693}a^{7}-\frac{1941197}{5212693}a^{6}+\frac{2533077}{5212693}a^{5}-\frac{186981}{5212693}a^{4}-\frac{258606}{5212693}a^{3}-\frac{149592}{306629}a^{2}-\frac{4534}{18037}a+\frac{171}{1061}$, $\frac{1}{10065710183}a^{39}-\frac{693407}{10065710183}a^{37}+\frac{845934}{10065710183}a^{36}-\frac{108418}{10065710183}a^{35}-\frac{538075}{10065710183}a^{34}+\frac{823782}{592100599}a^{33}-\frac{8140079}{10065710183}a^{32}+\frac{6634654}{10065710183}a^{31}+\frac{13194391}{10065710183}a^{30}-\frac{9296014}{10065710183}a^{29}+\frac{116898777}{10065710183}a^{28}-\frac{216174218}{10065710183}a^{27}-\frac{292366966}{10065710183}a^{26}+\frac{223105521}{10065710183}a^{25}+\frac{285773693}{10065710183}a^{24}+\frac{157464777}{10065710183}a^{23}-\frac{3696445}{592100599}a^{22}+\frac{115014856}{10065710183}a^{21}-\frac{28121373}{10065710183}a^{20}-\frac{166955380}{10065710183}a^{19}-\frac{276404228}{10065710183}a^{18}+\frac{212777769}{10065710183}a^{17}+\frac{8750953}{592100599}a^{16}-\frac{3650346423}{10065710183}a^{15}-\frac{2093441556}{10065710183}a^{14}-\frac{73884553}{592100599}a^{13}-\frac{5008118102}{10065710183}a^{12}-\frac{3191682121}{10065710183}a^{11}-\frac{1865189729}{10065710183}a^{10}+\frac{3881528539}{10065710183}a^{9}+\frac{3088065650}{10065710183}a^{8}-\frac{2380044768}{10065710183}a^{7}-\frac{931304273}{10065710183}a^{6}-\frac{3593236492}{10065710183}a^{5}+\frac{1763766231}{10065710183}a^{4}+\frac{4867707625}{10065710183}a^{3}+\frac{21345162}{592100599}a^{2}+\frac{4863152}{34829447}a+\frac{320040}{2048791}$, $\frac{1}{138775946293021}a^{40}-\frac{4824}{138775946293021}a^{39}+\frac{7277761}{138775946293021}a^{38}-\frac{9879772729}{138775946293021}a^{37}+\frac{9193626434}{138775946293021}a^{36}-\frac{30741187}{480193585789}a^{35}-\frac{7440172786}{138775946293021}a^{34}+\frac{22774559191}{138775946293021}a^{33}+\frac{14976882745}{138775946293021}a^{32}+\frac{119886313534}{138775946293021}a^{31}-\frac{54021005895}{138775946293021}a^{30}+\frac{212046587000}{138775946293021}a^{29}+\frac{262733580434}{138775946293021}a^{28}-\frac{127275818638}{138775946293021}a^{27}+\frac{2275994354817}{138775946293021}a^{26}-\frac{2522945137919}{138775946293021}a^{25}-\frac{970876768540}{138775946293021}a^{24}-\frac{1875168137693}{138775946293021}a^{23}-\frac{3450232024695}{138775946293021}a^{22}+\frac{430768055506}{138775946293021}a^{21}-\frac{288177305430}{138775946293021}a^{20}-\frac{2128228689425}{138775946293021}a^{19}+\frac{1957374771750}{138775946293021}a^{18}-\frac{130769782467}{8163290958413}a^{17}+\frac{3239512708677}{138775946293021}a^{16}-\frac{39047292592520}{138775946293021}a^{15}-\frac{30828516016660}{138775946293021}a^{14}-\frac{29013732515668}{138775946293021}a^{13}-\frac{51009137132875}{138775946293021}a^{12}+\frac{22048106632077}{138775946293021}a^{11}+\frac{12320459194338}{138775946293021}a^{10}+\frac{31621193366341}{138775946293021}a^{9}+\frac{43016344133621}{138775946293021}a^{8}-\frac{4214504147242}{138775946293021}a^{7}+\frac{33505348223429}{138775946293021}a^{6}-\frac{57632701720672}{138775946293021}a^{5}+\frac{983379550071}{138775946293021}a^{4}-\frac{1750453137359}{8163290958413}a^{3}+\frac{175215609328}{480193585789}a^{2}+\frac{13273729004}{28246681517}a+\frac{87947518}{1661569501}$, $\frac{1}{78\!\cdots\!99}a^{41}+\frac{20\!\cdots\!71}{78\!\cdots\!99}a^{40}+\frac{17\!\cdots\!65}{46\!\cdots\!47}a^{39}-\frac{41\!\cdots\!80}{78\!\cdots\!99}a^{38}-\frac{64\!\cdots\!32}{78\!\cdots\!99}a^{37}-\frac{51\!\cdots\!92}{78\!\cdots\!99}a^{36}-\frac{54\!\cdots\!79}{78\!\cdots\!99}a^{35}+\frac{53\!\cdots\!54}{78\!\cdots\!99}a^{34}+\frac{99\!\cdots\!48}{78\!\cdots\!99}a^{33}+\frac{34\!\cdots\!67}{78\!\cdots\!99}a^{32}-\frac{27\!\cdots\!39}{78\!\cdots\!99}a^{31}+\frac{11\!\cdots\!84}{78\!\cdots\!99}a^{30}+\frac{11\!\cdots\!66}{78\!\cdots\!99}a^{29}+\frac{70\!\cdots\!05}{78\!\cdots\!99}a^{28}-\frac{22\!\cdots\!43}{78\!\cdots\!99}a^{27}-\frac{14\!\cdots\!59}{78\!\cdots\!99}a^{26}+\frac{17\!\cdots\!62}{78\!\cdots\!99}a^{25}+\frac{51\!\cdots\!78}{78\!\cdots\!99}a^{24}+\frac{14\!\cdots\!16}{78\!\cdots\!99}a^{23}+\frac{15\!\cdots\!06}{78\!\cdots\!99}a^{22}-\frac{19\!\cdots\!83}{78\!\cdots\!99}a^{21}-\frac{18\!\cdots\!87}{78\!\cdots\!99}a^{20}+\frac{16\!\cdots\!86}{78\!\cdots\!99}a^{19}+\frac{10\!\cdots\!26}{78\!\cdots\!99}a^{18}+\frac{46\!\cdots\!97}{78\!\cdots\!99}a^{17}+\frac{29\!\cdots\!80}{46\!\cdots\!47}a^{16}-\frac{25\!\cdots\!70}{78\!\cdots\!99}a^{15}+\frac{16\!\cdots\!20}{78\!\cdots\!99}a^{14}-\frac{33\!\cdots\!44}{78\!\cdots\!99}a^{13}-\frac{39\!\cdots\!15}{78\!\cdots\!99}a^{12}+\frac{12\!\cdots\!36}{78\!\cdots\!99}a^{11}+\frac{14\!\cdots\!05}{78\!\cdots\!99}a^{10}-\frac{52\!\cdots\!83}{46\!\cdots\!47}a^{9}-\frac{38\!\cdots\!93}{78\!\cdots\!99}a^{8}+\frac{35\!\cdots\!43}{78\!\cdots\!99}a^{7}-\frac{11\!\cdots\!78}{78\!\cdots\!99}a^{6}+\frac{10\!\cdots\!27}{78\!\cdots\!99}a^{5}-\frac{20\!\cdots\!21}{78\!\cdots\!99}a^{4}-\frac{22\!\cdots\!50}{46\!\cdots\!47}a^{3}+\frac{94\!\cdots\!11}{27\!\cdots\!91}a^{2}+\frac{23\!\cdots\!12}{15\!\cdots\!23}a+\frac{29\!\cdots\!21}{93\!\cdots\!19}$ Copy content Toggle raw display

sage: K.integral_basis()
 
gp: K.zk
 
magma: IntegralBasis(K);
 
oscar: basis(OK)
 

Monogenic:  No
Index:  Not computed
Inessential primes:  $17$

Class group and class number

not computed

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 
oscar: class_group(K)
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, fUK := UnitGroup(K);
 
oscar: UK, fUK = unit_group(OK)
 
Rank:  $20$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( -1 \)  (order $2$) Copy content Toggle raw display
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
oscar: torsion_units_generator(OK)
 
Fundamental units:  not computed
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K|fUK(g): g in Generators(UK)];
 
oscar: [K(fUK(a)) for a in gens(UK)]
 
Regulator:  not computed
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr $ not computed \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^42 + 6*x^40 - 4*x^39 - 207*x^38 - 72*x^37 - 857*x^36 + 225*x^35 + 13524*x^34 + 4498*x^33 + 47646*x^32 - 1779*x^31 - 319728*x^30 + 56511*x^29 - 873558*x^28 + 1268358*x^27 + 3182931*x^26 - 287550*x^25 + 12585242*x^24 - 25238916*x^23 + 37767705*x^22 + 35532417*x^21 + 14671665*x^20 + 387146580*x^19 - 33604015*x^18 + 83426931*x^17 + 893018691*x^16 - 978644267*x^15 + 2536898589*x^14 + 320653281*x^13 + 1627927205*x^12 + 1198131426*x^11 - 1595121042*x^10 - 2227675498*x^9 - 1701355491*x^8 - 2415610293*x^7 - 905477771*x^6 + 53305794*x^5 + 1490523864*x^4 + 2070850257*x^3 + 1799361396*x^2 + 662562267*x + 198529417)
 
DK = K.disc(); r1,r2 = K.signature(); RK = K.regulator(); RR = RK.parent()
 
hK = K.class_number(); wK = K.unit_group().torsion_generator().order();
 
2^r1 * (2*RR(pi))^r2 * RK * hK / (wK * RR(sqrt(abs(DK))))
 
# self-contained Pari/GP code snippet to compute the analytic class number formula
 
K = bnfinit(x^42 + 6*x^40 - 4*x^39 - 207*x^38 - 72*x^37 - 857*x^36 + 225*x^35 + 13524*x^34 + 4498*x^33 + 47646*x^32 - 1779*x^31 - 319728*x^30 + 56511*x^29 - 873558*x^28 + 1268358*x^27 + 3182931*x^26 - 287550*x^25 + 12585242*x^24 - 25238916*x^23 + 37767705*x^22 + 35532417*x^21 + 14671665*x^20 + 387146580*x^19 - 33604015*x^18 + 83426931*x^17 + 893018691*x^16 - 978644267*x^15 + 2536898589*x^14 + 320653281*x^13 + 1627927205*x^12 + 1198131426*x^11 - 1595121042*x^10 - 2227675498*x^9 - 1701355491*x^8 - 2415610293*x^7 - 905477771*x^6 + 53305794*x^5 + 1490523864*x^4 + 2070850257*x^3 + 1799361396*x^2 + 662562267*x + 198529417, 1);
 
[polcoeff (lfunrootres (lfuncreate (K))[1][1][2], -1), 2^K.r1 * (2*Pi)^K.r2 * K.reg * K.no / (K.tu[1] * sqrt (abs (K.disc)))]
 
/* self-contained Magma code snippet to compute the analytic class number formula */
 
Qx<x> := PolynomialRing(QQ); K<a> := NumberField(x^42 + 6*x^40 - 4*x^39 - 207*x^38 - 72*x^37 - 857*x^36 + 225*x^35 + 13524*x^34 + 4498*x^33 + 47646*x^32 - 1779*x^31 - 319728*x^30 + 56511*x^29 - 873558*x^28 + 1268358*x^27 + 3182931*x^26 - 287550*x^25 + 12585242*x^24 - 25238916*x^23 + 37767705*x^22 + 35532417*x^21 + 14671665*x^20 + 387146580*x^19 - 33604015*x^18 + 83426931*x^17 + 893018691*x^16 - 978644267*x^15 + 2536898589*x^14 + 320653281*x^13 + 1627927205*x^12 + 1198131426*x^11 - 1595121042*x^10 - 2227675498*x^9 - 1701355491*x^8 - 2415610293*x^7 - 905477771*x^6 + 53305794*x^5 + 1490523864*x^4 + 2070850257*x^3 + 1799361396*x^2 + 662562267*x + 198529417);
 
OK := Integers(K); DK := Discriminant(OK);
 
UK, fUK := UnitGroup(OK); clK, fclK := ClassGroup(OK);
 
r1,r2 := Signature(K); RK := Regulator(K); RR := Parent(RK);
 
hK := #clK; wK := #TorsionSubgroup(UK);
 
2^r1 * (2*Pi(RR))^r2 * RK * hK / (wK * Sqrt(RR!Abs(DK)));
 
# self-contained Oscar code snippet to compute the analytic class number formula
 
Qx, x = PolynomialRing(QQ); K, a = NumberField(x^42 + 6*x^40 - 4*x^39 - 207*x^38 - 72*x^37 - 857*x^36 + 225*x^35 + 13524*x^34 + 4498*x^33 + 47646*x^32 - 1779*x^31 - 319728*x^30 + 56511*x^29 - 873558*x^28 + 1268358*x^27 + 3182931*x^26 - 287550*x^25 + 12585242*x^24 - 25238916*x^23 + 37767705*x^22 + 35532417*x^21 + 14671665*x^20 + 387146580*x^19 - 33604015*x^18 + 83426931*x^17 + 893018691*x^16 - 978644267*x^15 + 2536898589*x^14 + 320653281*x^13 + 1627927205*x^12 + 1198131426*x^11 - 1595121042*x^10 - 2227675498*x^9 - 1701355491*x^8 - 2415610293*x^7 - 905477771*x^6 + 53305794*x^5 + 1490523864*x^4 + 2070850257*x^3 + 1799361396*x^2 + 662562267*x + 198529417);
 
OK = ring_of_integers(K); DK = discriminant(OK);
 
UK, fUK = unit_group(OK); clK, fclK = class_group(OK);
 
r1,r2 = signature(K); RK = regulator(K); RR = parent(RK);
 
hK = order(clK); wK = torsion_units_order(K);
 
2^r1 * (2*pi)^r2 * RK * hK / (wK * sqrt(RR(abs(DK))))
 

Galois group

$C_{42}$ (as 42T1):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: G = GaloisGroup(K);
 
oscar: G, Gtx = galois_group(K); G, transitive_group_identification(G)
 
A cyclic group of order 42
The 42 conjugacy class representatives for $C_{42}$
Character table for $C_{42}$ is not computed

Intermediate fields

\(\Q(\sqrt{-87}) \), \(\Q(\zeta_{9})^+\), 6.0.480048687.1, 7.7.594823321.1, 14.0.22439994995240462987343.1, 21.21.4814587615056751193058435502319478353721.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

sage: K.subfields()[1:-1]
 
gp: L = nfsubfields(K); L[2..length(b)]
 
magma: L := Subfields(K); L[2..#L];
 
oscar: subfields(K)[2:end-1]
 

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type $21^{2}$ R $42$ $21^{2}$ $21^{2}$ $21^{2}$ ${\href{/padicField/17.1.0.1}{1} }^{42}$ ${\href{/padicField/19.14.0.1}{14} }^{3}$ $42$ R $42$ ${\href{/padicField/37.14.0.1}{14} }^{3}$ ${\href{/padicField/41.3.0.1}{3} }^{14}$ $42$ $21^{2}$ ${\href{/padicField/53.14.0.1}{14} }^{3}$ ${\href{/padicField/59.6.0.1}{6} }^{7}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Sage:
 
p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
\\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Pari:
 
p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
 
// to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7 in Magma:
 
p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
 
# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Oscar:
 
p = 7; pfac = factor(ideal(ring_of_integers(K), p)); [(e, valuation(norm(pr),p)) for (pr,e) in pfac]
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(3\) Copy content Toggle raw display Deg $42$$6$$7$$63$
\(29\) Copy content Toggle raw display Deg $42$$14$$3$$39$